Dimension reduction for functional data based on weak conditional moments
提出一种新的函数型线性充分降维方法，基于弱条件期望（Carleman算子诱导）替代传统条件均值/方差估计，避免函数空间中的维数灾难，适用于预测变量和响应均为随机函数的情形。
We develop a general theory and estimation methods for functional linear sufficient dimension reduction, where both the predictor and the response can be random functions, or even vectors of functions. Unlike the existing dimension reduction methods, our approach does not rely on the estimation of conditional mean and conditional variance. Instead, it is based on a new statistical construction—the weak conditional expectation, which is based on Carleman operators and their inducing functions. Weak conditional expectation is a generalization of conditional expectation. Its key advantage is to replace the projection on to an L2-space—which defines conditional expectation—by projection on to an arbitrary Hilbert space, while still maintaining the unbiasedness of the related dimension reduction methods. This flexibility is particularly important for functional data, because attempting to estimate a full-fledged conditional mean or conditional variance by slicing or smoothing over the space of vector-valued functions may be inefficient due to the curse of dimensionality. We evaluated the performances of the our new methods by simulation and in several applied settings.